class: center, middle, inverse, title-slide # Panel data models ## Tutorial 2 ### Stanislav Avdeev --- # Panel data - Panel data is when you observe the same individual over multiple time periods - "Individual" could be a person, a company, a state, a country, etc. There are `\(N\)` individuals in the panel data - "Time period" could be a year, a month, a day, etc. There are `\(T\)` time periods in the data - We assume that we observe each individual the same number of times, i.e. a *balanced* panel (so we have `\(N\times T\)` observations) - You can use these estimators with unbalanced panels too, it just gets a little more complex --- # Panel data - Here's what (a few rows from) a panel data set looks like - a variable for individual (county), a variable for time (year), and then the data <table> <thead> <tr> <th style="text-align:right;"> County </th> <th style="text-align:right;"> Year </th> <th style="text-align:right;"> CrimeRate </th> <th style="text-align:right;"> ProbofArrest </th> </tr> </thead> <tbody> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 81 </td> <td style="text-align:right;"> 0.0398849 </td> <td style="text-align:right;"> 0.289696 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 82 </td> <td style="text-align:right;"> 0.0383449 </td> <td style="text-align:right;"> 0.338111 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 83 </td> <td style="text-align:right;"> 0.0303048 </td> <td style="text-align:right;"> 0.330449 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 84 </td> <td style="text-align:right;"> 0.0347259 </td> <td style="text-align:right;"> 0.362525 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 85 </td> <td style="text-align:right;"> 0.0365730 </td> <td style="text-align:right;"> 0.325395 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 86 </td> <td style="text-align:right;"> 0.0347524 </td> <td style="text-align:right;"> 0.326062 </td> </tr> <tr> <td style="text-align:right;"> 1 </td> <td style="text-align:right;"> 87 </td> <td style="text-align:right;"> 0.0356036 </td> <td style="text-align:right;"> 0.298270 </td> </tr> <tr> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 81 </td> <td style="text-align:right;"> 0.0163921 </td> <td style="text-align:right;"> 0.202899 </td> </tr> <tr> <td style="text-align:right;"> 3 </td> <td style="text-align:right;"> 82 </td> <td style="text-align:right;"> 0.0190651 </td> <td style="text-align:right;"> 0.162218 </td> </tr> </tbody> <tfoot> <tr> <td style = 'padding: 0; border:0;' colspan='100%'><sup></sup> 9 rows out of 630. "Prob. of Arrest" is estimated probability of being arrested when you commit a crime</td> </tr> </tfoot> </table> --- # Between and within variation - Let's pick a few counties and graph this out <!-- --> --- # Between and within variation - If we look at the overall variation by using the pooled OLS estimator, we get this <!-- --> --- # Between and within variation - **Between** variation is what we get if we look at the relationship between the *means of each county* <!-- --> --- # Between and within variation - The individual year-to-year variation within county doesn't matter <!-- --> --- # Between and within variation - Within variation goes the other way - it treats those orange crosses as their own individualized sets of axes and looks at variation *within* county from year-to-year only! - We basically slide the crosses over on top of each other and then analyze *that* data <!-- --> --- # Between and within variation - We can clearly see that *between counties* there's a strong positive relationship - But if you look *within* a given county, the relationship isn't that strong, and actually seems to be negative - Which would make sense - if you think your chances of getting arrested are high, that should be a deterrent to crime - We are ignoring all differences between counties and looking only at differences within counties - Fixed effects is sometimes also referred to as the “within” estimator --- # Panel data model The `\(it\)` subscript says this variable varies over individual `\(i\)` and time `\(t\)` `$$Y_{it} = \alpha + X_{it}' \beta + U_{it}$$` - What if there are individual-level components in the error term causing omitted variable bias? - `\(X_{it}\)` might be related to the variable which is not in the model and thus in the error term - Regular omitted variable bias. If we don't adjust for the individual effect, we get a biased `\(\hat{\beta}\)` - So we really have this then: `$$Y_{it} = \alpha + X_{it}' \beta + \eta_i + U_{it}$$` --- # Fixed effects: estimation - To estimate fixed effects model, we need to remove between variation so that all that's left is within variation - There are two main ways: *de-meaning* and *binary variables* (they give the same result, for balanced panels anyway) - Let's do de-meaning first, since it's most closely and obviously related to the "removing between variation" explanation we've been going for - The process here is simple 1. For each variable `\(X_{it}\)`, `\(Y_{it}\)`, etc., get the mean value of that variable for each individual `\(\bar{X}_i, \bar{Y}_i\)` 2. Subtract out that mean to get residuals `\((X_{it} - \bar{X}_i), (Y_{it} - \bar{Y}_i)\)` 3. Work with those residuals - That `\(\alpha\)` and `\(\eta_u\)` terms get absorbed - The residuals are, by construction, no longer related to the `\(\alpha_i\)`, so it no longer goes in the residuals `$$Y_{it} - \bar{Y}_i = (X_{it} - \bar{X}_i)' \beta + (U_{it} - \bar{U_{i}})$$` --- # De-meaning: empirical application - We can use `group_by` to get means-within-groups and subtract them out ```r data(crime4, package = 'wooldridge') crime4 <- crime4 %>% # Filter to the data points from our graph filter(county %in% c(1,3,7, 23), prbarr < .5) %>% group_by(county) %>% mutate(mean_crime = mean(crmrte), mean_prob = mean(prbarr)) %>% mutate(demeaned_crime = crmrte - mean_crime, demeaned_prob = prbarr - mean_prob) ``` --- # De-meaning: empirical application ```r orig_data <- lm(crmrte ~ prbarr, data = crime4) de_mean <- lm(demeaned_crime ~ demeaned_prob, data = crime4) export_summs(orig_data, de_mean) ``` <table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto; " id="tab:unnamed-chunk-8"> <col><col><col><tr> <th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 1</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 2</th></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(Intercept)</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.01 * </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.00 </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.00) </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">prbarr</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.05 **</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.02) </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">demeaned_prob</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.03 *</td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R2</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.25 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.21 </td></tr> <tr> <th colspan="3" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001; ** p < 0.01; * p < 0.05.</th></tr> </table> --- # De-meaning approach: interpreting a within relationship - How can we interpret that slope of `-0.03`? - This is all *within variation* so our interpretation must be *within-county* - If we think we've causally identified it, it means that "raising the arrest probability by 1 percentage point in a county reduces the number of crimes per person in that county by .0003". - We're basically "controlling for county" - So your interpretation should think of it in that way - *holding county constant* i.e. *comparing two observations with the same value of county* i.e. *comparing a county to itself at a different point in time* --- # Least squares dummy variable (LSDV) approach - De-meaning the data is not the only way to do it - And sometimes it can make the standard errors wonky, since they don't recognize that you've estimated those means - You can also use the least squares dummy variable (another word for "binary variable") method - We just treat "individual" like the categorical variable it is and add it as a control --- # LSDV approach: empirical application ```r lsdv <- lm(crmrte ~ prbarr + factor(county), data = crime4) export_summs(orig_data, de_mean, lsdv, coefs = c('prbarr', 'demeaned_prob')) ``` <table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto; " id="tab:unnamed-chunk-9"> <col><col><col><col><tr> <th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 1</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 2</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 3</th></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">prbarr</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.05 **</td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.03 *</td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.02) </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">demeaned_prob</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.03 *</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R2</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.25 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.21 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.94 </td></tr> <tr> <th colspan="4" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001; ** p < 0.01; * p < 0.05.</th></tr> </table> --- # LSDV approach: interpretation - The result is the same, as it should be - Except for that `\(R^2\)` - why is it so much higher for LSDV? - Because de-meaning takes out the part explained by the fixed effects ( `\(\alpha_i\)` ) *before* running the regression, while LSDV does it *in* the regression - So the .94 is the portion of `crmrte` explained by `prbarr` *and* `county`, whereas the .21 is the "within - `\(R^2\)` " - the portion of *the within variation* that's explained by `prbarr` - Neither is wrong (and the .94 isn't "better"), they're just measuring different things --- # LSDV approach: interpretation - A benefit of the LSDV approach is that it calculates the fixed effects `\(\alpha_i\)` for you - We left those out of the table with the `coefs` argument of `export_summs` (we rarely want them) but here they are: ``` ## ## Call: ## lm(formula = crmrte ~ prbarr + factor(county), data = crime4) ## ## Coefficients: ## (Intercept) prbarr factor(county)3 factor(county)7 ## 0.045631 -0.030491 -0.025308 -0.009870 ## factor(county)23 ## -0.008587 ``` - Interpretation is exactly the same as with a categorical variable - we have an omitted county, and these show the difference relative to that omitted county --- # LSDV approach: interpretation - This also makes clear another element of what's happening. Just like with a categorical variable, the line is moving *up and down* to meet the counties - Graphically, de-meaning moves all the points together in the middle to draw a line, while LSDV moves the line up and down to meet the points <!-- --> --- # LSDV approach: interpretation - LSDV is computationally expensive - If there are a lot of individuals, or big data, or if you have many sets of fixed effects, it can be very slow - Most professionally made fixed-effects commands use de-meaning, but then adjust the standard errors properly - They also leave the fixed effects coefficients off the regression table by default --- # Panel data: estimation - Applied researchers rarely do either of these, and rather will use a command specifically designed for fixed effects - In R, there are three big ones: `feols` in **fixest**, `felm` in **lfe**, `plm` in **plm**, or or `lm_robust` in **estimatr** - `feols` seems to be a better choice. - **fixest** does all sorts of other neat stuff like fixed effects in nonlinear models like logit, regression tables, joint-test functions, and on and on - It’s very fast, and can be easily adjusted to do FE with other regression methods like logit, or combined with instrumental variables - It clusters the standard errors by the first fixed effect by default, which we usually want --- # Panel data: estimation ```r pro <- feols(crmrte ~ prbarr | county, data = crime4) export_summs(de_mean, pro, statistics = c(N = 'nobs', R2 = 'r.squared')) ``` <table class="huxtable" style="border-collapse: collapse; border: 0px; margin-bottom: 2em; margin-top: 2em; ; margin-left: auto; margin-right: auto; " id="tab:unnamed-chunk-12"> <col><col><col><tr> <th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 1</th><th style="vertical-align: top; text-align: center; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">Model 2</th></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(Intercept)</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.00 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.00) </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">demeaned_prob</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.03 *</td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">prbarr</th><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">-0.03 *</td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"></th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.4pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">(0.01) </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">N</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0.4pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">27 </td></tr> <tr> <th style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">R2</th><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.21 </td><td style="vertical-align: top; text-align: right; white-space: normal; border-style: solid solid solid solid; border-width: 0pt 0pt 0.8pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;">0.94 </td></tr> <tr> <th colspan="3" style="vertical-align: top; text-align: left; white-space: normal; border-style: solid solid solid solid; border-width: 0.8pt 0pt 0pt 0pt; padding: 6pt 6pt 6pt 6pt; font-weight: normal;"> *** p < 0.001; ** p < 0.01; * p < 0.05.</th></tr> </table> --- # Limits to fixed effects - Remember we are isolating within variation - If an individual has no within variation, say their treatment never changes, they basically get washed out entirely! - A fixed-effects regression wouldn’t represent them. And can’t use FE to study things that are fixed over time - And in general if there’s not a lot of within variation, FE is going to be very noisy. Make sure there’s variation to study! --- # Limits to fixed effects 1. They don't control for anything that has within variation 2. They control away *everything* that's between-only, so we can't see the effect of anything that's between-only 3. Anything with only a *little* within variation will have most of its variation washed out too 4. The estimate pays the most attention to individuals with *lots of variation in treatment* - 2 and 3 can be addressed by using "random effects" instead --- # Inference - It’s common to cluster standard errors at the level of the fixed effects, since it seems likely that errors would be correlated over time (autocorrelated errors) - It’s possible to have more than one set of fixed effects - But interpretation gets tricky - think through what variation in X you’re looking at at that point --- # References * Slides